Common Fixed Point Theorems for Weakly Compatible Mappings in Fuzzy Metric Spaces Using (JCLR) Property ()
1. Introduction
In 1965, Zadeh [1] investigated the concept of a fuzzy set in his seminal paper. In the last two decades there has been a tremendous development and growth in fuzzy mathematics. The concept of fuzzy metric space was introduced by Kramosil and Michalek [2] in 1975, which opened an avenue for further development of analysis in such spaces. Further, George and Veeramani [3] modified the concept of fuzzy metric space introduced by Kramosil and Michalek [2] with a view to obtain a Hausdoroff topology which has very important applications in quantum particle physics, particularly in connection with both string and 
theory (see, [4] and references mentioned therein). Fuzzy set theory also has applications in applied sciences such as neural network theory, stability theory, mathematical programming, modeling theory, engineering sciences, medical sciences (medical genetics, nervous system), image processing, control theory, communication etc.
In 2002, Aamri and El-Moutawakil [5] defined the notion of (E.A) property for self mappings which contained the class of non-compatible mappings in metric spaces. It was pointed out that (E.A) property allows replacing the completeness requirement of the space with a more natural condition of closedness of the range as well as relaxes the complexness of the whole space, continuity of one or more mappings and containment of the range of one mapping into the range of other which is utilized to construct the sequence of joint iterates. Subsequently, there are a number of results proved for contraction mappings satisfying (E.A) property in fuzzy metric spaces (see [6-11]). Most recently, Sintunavarat and Kumam [12] defined the notion of “common limit in the range” property (or (CLR) property) in fuzzy metric spaces and improved the results of Mihet [10]. In [12], it is observed that the notion of (CLR) property never requires the condition of the closedness of the subspace while (E.A) property requires this condition for the existence of the fixed point (also see [13]). Many authors have proved common fixed point theorems in fuzzy metric spaces for different contractive conditions. For details, we refer to [14-25].
The aim of this paper is to introduce the notion of the joint common limit in the range of mappings property called (JCLR) property and prove a common fixed point theorem for a pair of weakly compatible mappings using (JCLR) property in fuzzy metric space. As an application to our main result, we present a common fixed point theorem for two finite families of self mappings in fuzzy metric space using the notion of pairwise commuting due to Imdad et al. [15]. Our results improve and generalize the results of Cho et al. [26], Abbas et al. [7] and Kumar [8].
2. Preliminaries
Definition 2.1 [27] A binary operation
is a continuous t-norm if it satisfies the following conditions:
1) 
is associative and commutative
2) 
is continuous
3) 
for all
4) 
whenever 
and 
for all
.
Examples of continuous t-norms are 
and
.
Definition 2.2 [3] A 3-tuple 
is said to be a fuzzy metric space if X is an arbitrary set, * is a continuous t-norm and M is a fuzzy set on 
satisfying the following conditions: For all
, 
,
1)
2) 
if and only if
3)
4)
5) 
is continuous.
Then M is called a fuzzy metric on X and 
denotes the degree of nearness between x and y with respect to t.
Let 
be a fuzzy metric space. For
, the open ball 
with center 
and radius 
is defined by
Now let 
be a fuzzy metric space and 
the set of all 
with 
if and only if there exist 
and 
such that
. Then 
is a topology on X induced by the fuzzy metric M.
In the following example (see [3]), we know that every metric induces a fuzzy metric:
Example 2.1 Let 
be a metric space. Denote 
(or
) for all 
and let 
be fuzzy sets on 
defined as follows:
.
Then 
is a fuzzy metric space and the fuzzy metric M induced by the metric d is often referred to as the standard fuzzy metric.
Definition 2.3 Let 
be a fuzzy metric space. M is said to be continuous on 
if
whenever a sequence 
in 
converge to a point
, i.e.,
and
Lemma 2.1 [28] Let 
be a fuzzy metric space. Then 
is non-decreasing for all
.
Lemma 2.2 [29] Let 
be a fuzzy metric space. If there exists 
such that
for all 
and
, then
.
Definition 2.4 [30] Two self mappings f and g of a non-empty set X are said to be weakly compatible (or coincidentally commuting) if they commute at their coincidence points, i.e. if 
some
, then
.
Remark 2.1 [30] Two compatible self mappings are weakly compatible, but the converse is not true. Therefore the concept of weak compatibility is more general than that of compatibility.
Definition 2.5 [7] A pair of self mappings f and g of a fuzzy metric space 
are said to satisfy the (E.A) property, if there exists a sequence 
in X for some 
such that
Remark 2.2 It is noted that weak compatibility and (E.A) property are independent to each other (see [31], Example 2.1, Example 2.2).
In 2011, Sintunavarat and Kumam [12] defined the notion of “common limit in the range” property in fuzzy metric space as follows:
Definition 2.6 A pair 
of self mappings of a fuzzy metric space 
is said to satisfy the “common limit in the range of g” property (shortly, (CLRg) property) if there exists a sequence 
in X such that
for some
.
Now, we show examples of self mappings f and g which are satisfying the (CLRg) property.
Example 2.2 Let 
be a fuzzy metric space with 
and
for all
. Define self mappings f and g on X by 
and 
for all
. Let a sequence 
in X, we have
which shows that f and g satisfy the (CLRg) property.
Example 2.3 The conclusion of Example 2.2 remains true if the self mappings f and g is defined on X by
and 
for all
. Let a sequence 
in X. Since
therefore f and g satisfy the (CLRg) property.
The following definition is on the lines due to Imdad et al. [32].
Definition 2.7 [32] Two families of self mappings
and 
are said to be pairwise commuting if
1) 
for all
2) 
for all
3) 
for all 
and 
.
Throughout this paper, 
is considered to be a fuzzy metric space with condition
for all
.
3. Main Results
In this section, we first introduce the notion of “the joint common limit in the range property” of two pairs of self mappings.
Definition 3.1 Let 
be a fuzzy metric space and
. The pair 
and 
are said to satisfy the “joint common limit in the range of b and g” property (shortly, (JCLRbg) property) if there exists a sequence 
and 
in X such that
(1)
for some
.
Remark 3.1 If
, 
and 
in (1), then we get the definition of (CLRg).
Throughout this section, 
denotes the set of all continuous and increasing functions 
in any coordinate and 
for all
.
Following are examples of some function
:
1) 
for some
.
2) 
for some
.
3) 
for some 
and for all t-norm 
such that
.
Now, we state and prove main results in this paper.
Theorem 3.1 Let 
be a fuzzy metric space, where 
is a continuous t-norm and f, g, a and b be mappings from X into itself. Further, let the pair 
and 
are weakly compatible and there exists a constant 
such that
(2)
holds for all
, 
, 
and
. If 
and 
satisfy the (JCLRbg) property, then f, g, a and b have a unique common fixed point in X.
Proof. Since the pairs 
and 
satisfy the (JCLRbg) property, there exists a sequence 
and 
in X such that
for some
.
Now we assert that
. Using (2), with
, 
, for
, we get
Taking the limit as
, we have
Since 
is increasing in each of its coordinate and 
for all
, we get 
. By Lemma 2.2, we have
.
Next we show that
. Using (2), with
, 
, for
, we get
Taking the limit as
, we have
Since 
is increasing in each of its coordinate and 
for all
, we get
. By Lemma 2.2, we have
.
Now, we assume that
. Since the pair 
is weakly compatible, 
and then
. It follows from 
is weakly compatible, 
and hence 
.
We show that
. To prove this, using (2) with
, 
, for
, we get
and so
Since 
is increasing in each of its coordinate and 
for all
, 
, which implies that
. Hence
.
Next, we show that
. To prove this, using (2) with
, 
, for
, we get
and so
Since 
is increasing in each of its coordinate and 
for all
, 
, which implies that
. Hence
. Therefore, we conclude that 
this implies f, g, a and b have common fixed point that is a point z.
For uniqueness of common fixed point, we let w be another common fixed point of the mappings f, g, a and b. On using (2) with
, 
, for
, we have
and then
Since 
is increasing in each of its coordinate and 
for all
, 
, which implies that
. Therefore f, g, a and b have a unique a common fixed point.
Remark 3.2 From the result, it is asserted that (JCLRgb) property never requires any condition closedness of the subspace, continuity of one or more mappings and containment of ranges amongst involved mappings.
Remark 3.3 Theorem 3.1 improves and generalizes the results of Abbas et al. ([7], Theorem 2.1) and Kumar ([8], Theorem 2.3) without any requirement of containment amongst range sets of the involved mappings and closedness of the underlying subspace.
Remark 3.4 Since the condition of t-norm with 
for all 
is replaced by arbitrary continuous t-norm, Theorem 3.1 also improves the result of Cho et al. ([26], Theorem 3.1) without any requirement of completeness of the whole space, continuity of one or more mappings and containment of ranges amongst involved mappings.
Corollary 3.1 Let 
be a fuzzy metric space, where 
is a continuous t-norm and f, g, a and b be mappings from X into itself. Further, let the pair 
and 
are weakly compatible and there exists a constant 
such that
(3)
holds for all
, 
, 
and 
such that
. If 
and 
satisfy the (JCLRbg) property, then f, g, a and b have a unique common fixed point in X.
Proof. By Theorem 3.1, if we define
then the result follows.
Remark 3.5 Corollary 3.1 improves the result of Cho et al. ([26], Corollary 3.4) without any requirement of completeness of the whole space, continuity of one or more mappings and containment of ranges amongst involved mappings while the condition of t-norm 
for all 
is replaced by arbitrary continuous t-norm.
Corollary 3.2 Let 
be a fuzzy metric space, where 
is a continuous t-norm and f and g be mappings from X into itself. Further, let the pair 
is weakly compatible and there exists a constant 
such that
(4)
holds for all
, 
, 
and
. If 
satisfies the (CLRg) property, then f and g have a unique common fixed point in X.
Proof. Take 
and 
in Theorem 3.1, then we get the result.
Our next theorem is proved for a pair of weakly compatible mappings in fuzzy metric space 
using (E.A) property under additional condition closedness of the subspace.
Theorem 3.2 Let 
be a fuzzy metric space, where 
is a continuous t-norm. Further, let the pair 
of self mappings is weakly compatible satisfying inequality (4) of Corollary 3.2. If f and g satisfy the (E.A) property and the range of g is a closed subspace of X, then f and g have a unique common fixed point in X.
Proof. Since the pair 
satisfies the (E.A) property, there exists a sequence 
in X such that
for some
. It follows from 
being a closed subspace of X that there exists 
in which
. Therefore f and g satisfy the (CLRg) property. From Corollary 3.2, the result follows.
In what follows, we present some illustrative examples which demonstrate the validity of the hypotheses and degree of utility of our results.
Example 3.1 Let 
with the metric d defined by 
and for each 
define
for all
. Clearly 
be a fuzzy metric space with t-norm defined by 
for all
. Consider a function 
defined by
. Then we have
. Define the self mappings f and g on X by
and
Taking 
or
, it is clear that the pair 
satisfies the (CLRg) property since
It is noted that
. Thus, all the conditions of Corollary 3.2 are satisfied for a fixed constant 
and 2 is a unique common fixed point of the pair
. Also, all the involved mappings are even discontinuous at their unique common fixed point 2. Here, it may be pointed out that 
is not a closed subspace of X.
Example 3.2 In the setting of Example 3.1, replace the mapping g by the following, besides retaining the rest:
Taking 
or
, it is clear that the pair 
satisfies the (E.A) property since
It is noted that
. Thus, all the conditions of Theorem 3.2 are satisfied and 2 is a unique common fixed point of the mappings f and g. Notice that all the involved mappings are even discontinuous at their unique common fixed point 2. Here, it is worth noting that 
is a closed subspace of X.
Now, we utilize Definition 2.7 which is a natural extension of commutativity condition to two finite families of self mappings. Our next theorem extends Corollary 3.2 in the following sense:
Theorem 3.3 Let 
and 
be two finite families of self mappings in fuzzy metric space
, where 
is a continuous t-norm such that 
and 
which satisfy the inequalities (4) of Corollary 3.2. If the pair 
shares (CLRg) property, then f and g have a unique point of coincidence.
Moreover, 
and 
have a unique common fixed point provided the pair of families 
commutes pairwise, where 
and 
.
Proof. The proof of this theorem can be completed on the lines of Theorem 3.1 contained in Imdad et al. [15], hence details are avoided.
Putting 
and
in Theorem 3.3, we get the following result:
Corollary 3.3 Let f and g be two self mappings of a fuzzy metric space
, where 
is a continuous t-norm. Further, let the pair 
shares (CLRg) property. Then there exists a constant 
such that
holds for all
, 
, 
, 
and m and n are fixed positive integers, then f and g have a unique common fixed point provided the pair 
commutes pairwise.
Remark 3.6 Theorem 3.2, Theorem 3.3 and Corollary 3.3 can also be outlined in respect of Corollary 3.1.
Remark 3.7 Using Example 2.2, we can obtain several fixed point theorems in fuzzy metric spaces in respect of Theorems 3.2 and 3.3 and Corollaries 3.2, 3.1 and 3.3.
4. Acknowledgements
The authors would like to express their sincere thanks to Professor Mujahid Abbas for his paper [18]. The second author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST). This study was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission under the Computational Science and Engineering Research Cluster (CSEC Grant No. 55000613).
NOTES
#Corresponding authors.